Fast approximate inverse power-iteration algorithm for adaptive total least-squares FIR filtering

The presence of contaminating noises at both the input and the output of an finite-impulse-response (FIR) system constitutes a major impediment to unbiased parameter estimation. The total least-squares (TLS) method is known to be effective in achieving unbiased estimation. In this correspondence, we develop a fast recursive algorithm with a view to finding the TLS solution for adaptive FIR filtering. Given the fact that the TLS solution is obtainable via inverse power iteration, we introduce a novel but approximate inverse power iteration in combination with Galerkin method so that the TLS solution can be updated adaptively at a lower computational cost. We also take advantage of the regular form of the TLS solution to constrain the last element of the filter parameter vector to the negative one. We further reduce the computational complexity of the developed algorithm by making efficient computation of the fast gain vector defined in [11] and using rank-one update of the augmented autocorrelation matrix. The developed algorithm saves seven M MAD's (number of multiplies, divides, and square roots) when compared with the recursive TLS algorithm in [6]. Moreover, unlike the algorithms given in [6] and [12], the developed algorithm does not deal with the solution to a one-variable quadratic equation and it avoids square root operation. Therefore, it has the simpler structure and may be more easily implemented. We then make a careful investigation into global convergence of the developed algorithm. Simulation results are provided that clearly illustrate appealing performance of the developed algorithm, including its good long-term numerical stability.